Subcritical Connectivity and Some Exact Tail Exponents in High Dimensional Percolation

Abstract

In high dimensional percolation at parameter $p<p_c$, the one-arm probability ${\pi }_p(n)$ is known to decay exponentially on scale $(p_c-p{)}^{-1/2}$. We show the same statement for the ratio ${\pi }_p(n)/{\pi }_{p_c}(n)$, establishing a form of a hypothesis of scaling theory. As part of our study, we provide sharp estimates (with matching upper and lower bounds) for several quantities of interest at the critical probability $p_c$. These include the tail behavior of volumes of, and chemical distances within, spanning clusters, along with the scaling of the two-point function at "mesoscopic distance" from the boundary of half-spaces. As a corollary, we obtain the tightness of the number of spanning clusters of a diameter $n$ box on scale $n^{d-6}$; this result complements a lower bound of Aizenman.

63 pages, 6 figures